Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | 2−1+4×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C3×2−1+4) |
| Fitting | C7×2−1+4 ⋊ C2×C3 |
| Hall | D7×2−1+4 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×2−1+4 |
| Sylow | C7 ⋊ C2×C3×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C7 | C25×C3 |
| Fitting | C7×2−1+4 | C2×C3 |
| hypercenter | 2−1+4 | AGL(1,𝔽7) |
| hypocenter | C7 | C2×C3×2−1+4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
1588 conjugacy classes of subgroups; only the 822 normal subgroups are shown.
This group has 1588 conjugacy classes of subgroups — too many to draw. Only its 822 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
119 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#577, AGL(1,𝔽7)×#585, AGL(1,𝔽7)×#454 |
| maximal normal subgroup of | AGL(1,𝔽7)×#577, AGL(1,𝔽7)×#585, AGL(1,𝔽7)×#454 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11446);